A Unified Software Framework for Empirical Gramians
A Unified Software Framework for Empirical Gramians
By Christian Himpe, and Mario Ohlberger
Institute for Computational and Applied Mathematics at the University of Muenster (2013)
Abstract Paper

Christian  Himpe

University of Münster

Germany

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acc %runs the code and outputs results. acc(1) %runs the code and outputs results and plots. Code is compatible to Octave >= 3.6.3. For further info see paper.
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February 19, 2013
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Abstract
A common approach in model reduction is balanced truncation, which is based on gramian matrices classifiying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical gramians not only extended this concept to nonlinear systems, but also provide a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical gramians. The gramian types will be discussed and applied in a model reduction framework for multiple-input-multiple-output (MIMO) systems.
Himpe, C., and M. Ohlberger, "A Unified Software Framework for Empirical Gramians", Institute for Computational and Applied Mathematics at the University of Muenster.
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Christian Himpe also created these companion sites

A Unified Software Framework for Empirical Gramians
Abstract
A common approach in model reduction is balanced truncation, which is based on gramian matrices classifiying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical gramians not only extended this concept to nonlinear systems, but also provide a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical gramians. The gramian types will be discussed and applied in a model reduction framework for multiple-input-multiple-output (MIMO) systems.
Himpe, C., "A Unified Software Framework for Empirical Gramians", Institute for Computational and Applied Mathematics at the University of Muenster.
Authors: Himpe
Ohlberger
Coders: Himpe
Last update
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9999
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Cross-Gramian Based Combined State and Parameter Reduction
Abstract
An accepted model reduction technique is balanced truncation, by which negligible states of a linear system of ODEs are determined by balancing the systems controllability and observability gramian matrices. To be applicable for nonlinear system this method was enhanced through the empirical gramians, while the cross gramian conjoined both gramians into one gramian matrix. This work introduces the empirical cross gramian for square Multiple-Input-Multiple-Output systems as well as the (empirical) joint gramian. Based on the cross gramian, the joint gramian determines, in addition to the Hankel singular values, the parameter identifiability allowing a combined model reduction, concurrently reducing state and parameter spaces. Furthermore, a controllability and an observability based combined reduction method are presented and the usage of empirical gramians is extended to parameter reduction in (Bayesian) inverse problems. All methods presented are evaluated by numerical experiments.
Himpe, C., "Cross-Gramian Based Combined State and Parameter Reduction", WWU Muenster.
Authors: Himpe
Ohlberger
Coders: Himpe
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Other Companion Sites on same paper

A Unified Software Framework for Empirical Gramians
A Unified Software Framework for Empirical Gramians
Abstract
A common approach in model reduction is balanced truncation, which is based on gramian matrices classifiying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical gramians not only extended this concept to nonlinear systems, but also provide a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical gramians. The gramian types will be discussed and applied in a model reduction framework for multiple-input-multiple-output (MIMO) systems.
Himpe, C., "A Unified Software Framework for Empirical Gramians", Institute for Computational and Applied Mathematics at the University of Muenster.
Authors: Himpe
Ohlberger
Coders: Himpe
Last update
02/05/2013
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9999
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A Unified Software Framework for Empirical Gramians
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A common approach in model reduction is balanced truncation, which is based on gramian matrices classifiying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical gramians not only extended this concept to nonlinear systems, but also provide a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical gramians. The gramian types will be discussed and applied in a model reduction framework for multiple-input-multiple-output (MIMO) systems.
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An accepted model reduction technique is balanced truncation, by which negligible states of a linear system of ODEs are determined by balancing the systems controllability and observability gramian matrices. To be applicable for nonlinear system this method was enhanced through the empirical gramians, while the cross gramian conjoined both gramians into one gramian matrix. This work introduces the empirical cross gramian for square Multiple-Input-Multiple-Output systems as well as the (empirical) joint gramian. Based on the cross gramian, the joint gramian determines, in addition to the Hankel singular values, the parameter identifiability allowing a combined model reduction, concurrently reducing state and parameter spaces. Furthermore, a controllability and an observability based combined reduction method are presented and the usage of empirical gramians is extended to parameter reduction in (Bayesian) inverse problems. All methods presented are evaluated by numerical experiments.
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